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Theorem tpid3g 3644
Description: Closed theorem form of tpid3 3645. (Contributed by Alan Sare, 24-Oct-2011.)
Assertion
Ref Expression
tpid3g (𝐴𝐵𝐴 ∈ {𝐶, 𝐷, 𝐴})

Proof of Theorem tpid3g
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elisset 2703 . 2 (𝐴𝐵 → ∃𝑥 𝑥 = 𝐴)
2 3mix3 1153 . . . . . . 7 (𝑥 = 𝐴 → (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴))
32a1i 9 . . . . . 6 (𝐴𝐵 → (𝑥 = 𝐴 → (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴)))
4 abid 2128 . . . . . 6 (𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴)} ↔ (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴))
53, 4syl6ibr 161 . . . . 5 (𝐴𝐵 → (𝑥 = 𝐴𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴)}))
6 dftp2 3578 . . . . . 6 {𝐶, 𝐷, 𝐴} = {𝑥 ∣ (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴)}
76eleq2i 2207 . . . . 5 (𝑥 ∈ {𝐶, 𝐷, 𝐴} ↔ 𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴)})
85, 7syl6ibr 161 . . . 4 (𝐴𝐵 → (𝑥 = 𝐴𝑥 ∈ {𝐶, 𝐷, 𝐴}))
9 eleq1 2203 . . . 4 (𝑥 = 𝐴 → (𝑥 ∈ {𝐶, 𝐷, 𝐴} ↔ 𝐴 ∈ {𝐶, 𝐷, 𝐴}))
108, 9mpbidi 150 . . 3 (𝐴𝐵 → (𝑥 = 𝐴𝐴 ∈ {𝐶, 𝐷, 𝐴}))
1110exlimdv 1792 . 2 (𝐴𝐵 → (∃𝑥 𝑥 = 𝐴𝐴 ∈ {𝐶, 𝐷, 𝐴}))
121, 11mpd 13 1 (𝐴𝐵𝐴 ∈ {𝐶, 𝐷, 𝐴})
Colors of variables: wff set class
Syntax hints:  wi 4  w3o 962   = wceq 1332  wex 1469  wcel 1481  {cab 2126  {ctp 3532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-3or 964  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3078  df-sn 3536  df-pr 3537  df-tp 3538
This theorem is referenced by:  rngmulrg  12109  srngmulrd  12116  lmodscad  12127  ipsmulrd  12135  ipsipd  12138  topgrptsetd  12145
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